2. Parallel Lines & Transversals

3. Standards/Objectives: Standard 3: Students will learn and apply geometric concepts. Objectives: Prove and use results about parallel lines and transversals. Use properties of parallel lines to solve real-life problems, such as estimating the Earth’s circumference

4. Vocab Parallel Lines – Coplanar lines that do not intersect. Parallel Planes – Two planes that do not intersect. Skew Lines – Lines that do not intersect and are not coplanar. Transversal – A line that intersects two or more lines in a plane at different points.

5. Example 1 a)Name all planes that are parallel to plane AEF Plane BHG b) Name all segments that intersect AF. EF, GF, DA, and BA c) Name all segments that are skew to AD FG, GB, EH, EC, and CH AB C D E F G H

6. Postulate 15 Corresponding Angles Postulate If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. 1 2  1 ≅  2

7. Theorem 3.4 Alternate Interior Angles If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. 3 4  3 ≅  4

8. Theorem 3.5 Consecutive Interior Angles If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary. 5 6  5 +  6 = 180 °

9. Theorem 3.6 Alternate Exterior Angles If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent. 7 8  7 ≅  8

10. Theorem 3.7 Perpendicular Transversal If a transversal is perpendicular to one of the two parallel lines, then it is perpendicular to the other. j  k j h k

11. Example 1: Proving the Alternate Interior Angles Theorem Given: p ║ q Prove:  1 ≅  2 1 2 3

12. Example 2 Identify each type of pairs of angles. a.) 7 and < 3 corresponding b.) 8 and 2 alternate exterior l a b c 1 2 34 56 78 910 1112

13. c.) 4 and 11 Corresponding d.) 7 and 1 alternate exterior e.) 3 and 9 alternate interior f.) 7 and 10 consecutive interior Is it really that easy?!?!

14. Example 2: Using properties of parallel lines Given that m  5 = 65 °, find each measure. Tell which postulate or theorem you use. A. m  6B. m  7 C. m  8D. m  9 6 7 5 8 9

15. Solutions: a.m  6 = m  5 = 65 ° Vertical Angles Theorem b.m  7 = 180 ° - m  5 =115 ° Linear Pair postulate c.m  8 = m  5 = 65 ° Corresponding Angles Postulate d.m  9 = m  7 = 115 ° Alternate Exterior Angles Theorem

16. Ex. 3—Classifying Leaves BOTANY—Some plants are classified by the arrangement of the veins in their leaves. In the diagram below, j ║ k. What is m  1? 120 ° j k 1

17. Solution 1.m  1 + 120 ° = 180° 2.m  1 = 60 ° 1.Consecutive Interior angles Theorem 2.Subtraction POE

18. Ex: Find: m  1= m  2= m  3= m  4= m  5= m  6= x= 125 o 2 1 3 4 6 5 x+15 o

19. Ex: Find: m  1=55 ° m  2=125 ° m  3=55 ° m  4=125 ° m  5=55 ° m  6=125 ° x=40 ° 125 o 2 1 3 4 6 5 x+15 o

20. Ex. 4: Using properties of parallel lines Use the properties of parallel lines to find the value of x. 125 ° 4 (x + 15) °

21. Another example 1 2 X -10 X + 30 < 1 = X-10 find X when < 2= X+30 X-10 + X+30= 180 2X+ 20 = 180 -20 2X = 160 X = 80

22. Example < 1 = 55 find X when < 2= X+30 1 2 55 X + 30 55 = X + 30 Alternate Ext. <‘s Are = < 1 = < 2 25 = X - 30